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Syllabus Edition
First teaching 2021
Last exams 2024
Angles at Centre & Semicircles (CIE IGCSE Maths: Extended)
Revision Note
What are circle theorems?
- You will have learned a lot of angle facts for your GCSE, including angles in polygons and angles with parallel lines
- Circle Theorems deal with angle facts that occur when lines are drawn within and connected to a circle
What do I need to know?
- You must be familiar with the names of parts of a circle including radius, diameter, arc, sector, chord, segment and tangent
- To solve some problems you may need to use the angle facts you are already familiar with from triangles, polygons, and parallel lines
- You may also have to use the formulae for circumference and area, so ensure you’re familiar with them
- (C = πd)
- (A = πr2)
Angles at Centre & Circumference
Circle Theorem: The angle subtended by an arc at the centre is twice the angle at the circumference
- This is one of the most useful circle theorems and forms a basis for many other angle facts within circles
- In this theorem, the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the same arc
- It is an easy circle theorem to spot on a diagram
STEP 1
Find any two radii in the circle and follow them to the circumference
STEP 2
See if there are lines from those points going to any other point on the circumference - When using this theorem in an exam you must use the keywords
- The angle at the centre is twice the angle at the circumference
- This theorem can also happen when the ‘triangle parts’ overlap:
Circle theorem: The angle in a semicircle is a right angle
Examiner Tip
- Add anything you can to a diagram you have been given
- Mark any equal radii and write in any angles and lengths you can work out, even if they don’t seem relevant to the actual question
- For each angle you work out, try to assign an angle fact or circle theorem to it
- Questions often ask for “reasons” and the names/titles/phrases for each of these is exactly what they are after
- When asked to “give reasons” aim to quote an angle fact or circle theorem for every angle you find, not just one for the final answer
Worked example
Find the value of in the diagram below.
There are three radii in the diagram, mark these as equal length lines. Notice how they create two isosceles triangles.
Base angles in isosceles triangles are equal, so this means that the angle next to must be 60°.
Using the circle theorem "The angle at the centre subtended by an arc is twice the angle at the circumference", form an equation for .
Expand the brackets and solve the equation.
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